Geometry of the eigencurve at critical Eisenstein series of weight 2
Journal de Théorie des Nombres de Bordeaux, Tome 27 (2015) no. 1, pp. 183-197.

Dans cet article, nous montrons que la série d’Eisenstein critique de poids 2, E 2 crit p , définit un point lisse dans la courbe de Hecke (l), où l est un nombre premier différent de p. Nous montrons également que E 2 crit p ,ord l définit un point lisse dans la courbe de Hecke pleine full (l) et que le point défini par E 2 crit p ,ord l 1 ,ord l 2 est non lisse dans la courbe de Hecke pleine full (l 1 l 2 ). En outre, nous montrons que (l) est étale sur l’espace des poids au point défini par E 2 crit p . En conséquence, nous montrons que la conjecture d’abaissement du niveau de Paulin n’est pas valide pour E 2 crit p ,ord l .

In this paper we show that the critical Eisenstein series of weight 2, E 2 crit p , defines a smooth point in the eigencurve (l), where l is a prime different from p. We also show that E 2 crit p ,ord l defines a smooth point in the full eigencurve full (l) and E 2 crit p ,ord l 1 ,ord l 2 defines a non-smooth point in the full eigencurve full (l 1 l 2 ). Further, we show that (l) is étale over the weight space at the point defined by E 2 crit p . As a consequence, we show that level lowering conjecture of Paulin fails to hold at E 2 crit p ,ord l .

Reçu le : 2013-10-31
Révisé le : 2014-10-01
Accepté le : 2014-10-16
Publié le : 2015-05-21
DOI : https://doi.org/10.5802/jtnb.898
Classification : 11F85,  11F11,  11F80
@article{JTNB_2015__27_1_183_0,
     author = {Dipramit Majumdar},
     title = {Geometry of the eigencurve at critical Eisenstein series of weight 2},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {27},
     number = {1},
     year = {2015},
     pages = {183-197},
     doi = {10.5802/jtnb.898},
     mrnumber = {3346969},
     zbl = {06554402},
     language = {en},
     url = {jtnb.centre-mersenne.org/item/JTNB_2015__27_1_183_0/}
}
Dipramit Majumdar. Geometry of the eigencurve at critical Eisenstein series of weight 2. Journal de Théorie des Nombres de Bordeaux, Tome 27 (2015) no. 1, pp. 183-197. doi : 10.5802/jtnb.898. https://jtnb.centre-mersenne.org/item/JTNB_2015__27_1_183_0/

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